Sketch a graph which shows that y is directly proportional to x². - OCR - GCSE Maths - Question 10 - 2018 - Paper 6

To illustrate that y is directly proportional to x², we must understand that this relationship produces a quadratic function. The relationship can be written as:

$y = kx^2$

where k is a positive constant. The graph of this equation is a parabola that opens upwards. Here are the steps to sketch the graph:

1. Identify the origin: Start at the origin (0,0) since when x=0, y will also equal 0.
2. Add points: Choose positive values of x (e.g., 1, 2, 3) and calculate the corresponding y values:
   • For x = 1, y = k(1²) = k
   • For x = 2, y = k(2²) = 4k
   • For x = 3, y = k(3²) = 9k
3. Plot points: Mark these points on the graph:
   • (1, k)
   • (2, 4k)
   • (3, 9k)
4. Draw the parabola: Connect these points with a smooth curve, which should illustrate the parabolic shape, confirming the direct proportionality.
5. Label axes: Ensure the x-axis and y-axis are appropriately labeled.
6. Choose and label k: You can select a specific value for k to illustrate the graph, often using k = 1 for simplicity.

This results in a clear graph showing the direct proportionality of y to x².